Numerical integration of double integral with two variables

Hi,

I am trying to numerically integrate the following double integral in MATLAB:

where Im is the imaginary part of expression, i is the imaginary number, x and y are variables while a, b, and c are constants.

Here is my attempt to solve this.

a = 3;
b = 4;
c = 5;
innerintegral = @(x) integral(@(y) (1i.*x.*y)./(y.^a-1i.*x),0,6);
outerintegral = integral(@(x) imag(exp(-1i.*x.*c+b.*innerintegral(x))),0,inf, 'ArrayValued', 1);

Is this the correct way of applying numerical integration with more than one variable? Also I am getting a warning when I run this expression which reads as "Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 6.7e+00. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy."

Can anyone please validate if the above implementation is a correct way of implementing the expressions state above.

Answer 1

[David Hill]

a = 3;
b = 4;
c = 5;
syms x y;
fun = @(x,y)(1i.*x.*y)./(y.^a-1i.*x);
I=vpaintegral(@(x)imag(exp(-1i.*x.*c+b.*vpaintegral(fun,y,0,6))),x,0,inf);

I get:

I= -0.0125514

[Shahrukh Khan Kasi]

Thank you David for your useful comment. I have executed the same expression with just one modification (shown in below picture) where I divide the whole expression with x. After doing so I am getting the error, "Failed precision goal. Try using 'MaxFunctionCalls'."

Do you think this is because we are using symbols in numerical integrations? Is there anyway to solve this? Also the execution of this script takes a very long to compute.

a = 3;
b = 0.0137;
c = 0.0023;
syms x y;
fun = @(x,y)(1i.*x.*y)./(y.^a-1i.*x);
I = vpaintegral(@(x)imag(exp(-1i.*x.*c+b.*vpaintegral(fun,y,0,6)))./x,x,0,inf);

Thanks.

[David Hill]

Runs fine with the following a,b,c constants. Not sure why the other constants are causing problems.

a = 3;
b = 4;
c = 5;
syms x y;
fun = @(x,y)(1i.*x.*y)./(y.^a-1i.*x);
I=vpaintegral(@(x)imag(exp(-1i.*x.*c+b.*vpaintegral(fun,y,0,6))./x),x,0,inf);

[Shahrukh Khan Kasi]

Yes it does with these set of dummy values but for different values of a, b and c, sometimes it runs fine while at other times it gives error for "Failed precision goal. Try using 'MaxFunctionCalls'." Also if I change the limits of first integral from (0,6) to (0,1) it gives the same error.

My question is should there be some additional check on the values of input constants such as a, b, c and the upper limit of inner integral due to the usage of vpaintegral? During the mathematical derivation of these expressions there is no such restriction on the values of a, b and c except that they have to be greater than 0. However with the numerical integration approach it run into "Failed precision goal."